Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian

Consider the Dirichlet-to-Neumann map $\Lambda_\beta$ associated with the Schr\"odinger operator $(D+\beta \A)^2$ with a magnetic potential in a bounded Lipschitz domain $\Omega$, where $\beta>1$ is the field strength parameter. Assume that the magnetic field $\B=\nabla \times \A$ is of finite type. We show that if $\beta>\beta_0$, the ground state for $\Lambda_\beta$ decays exponentially away from a neighborhood of the subset of $\partial\Omega$, on which $\B$ vanishes to the maximal order.